Furthermore, it was shown in [1]} that for some values \(r\)  and \(q\) , the bound (REF ) exceeds the asymptotic Gilbert-Varshamov bound for locally repairable codes.
Li et al. [2]} generalized the idea given in [1]} by considering more subgroups of automorphism groups of function fields in the Garcia-Stichtenoth tower [4]}. This construction allows more flexibility of locality. In particular, if \(r+1=up^v\)  with \(u|\gcd (p^v-1, \sqrt{q}-1)\) , then there exists a family of explicit \(q\) -ary linear locally repairable codes with locality \(r\)  whose rate \(R\)  and relative distance \(\delta \)  satisfy
\(R\geqslant \frac{r}{1+r}\Big {(}1-\delta -\frac{\sqrt{q}+r-1}{q-\sqrt{q}}\Big {)}.\) 
